Micromechanical Multiscale Fracture Model for Compressive Strength of Blended Cement Pastes

Cement and Concrete Research 83 (2016) 188–202

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Cement and Concrete Research

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Micromechanical multiscale fracture model for compressive strength of blended cement pastes

Michal Hlobil a,b,⁎,VítŠmilauer a, Gilles Chanvillard c,1 a Department of Mechanics, Faculty of Civil Engineering, CTU in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic b Institute for Mechanics of Materials and Structures (IMWS), Vienna University of Technology (TU Wien), Karlsplatz 13/202, A-1040 Wien, Austria c Lafarge Centre de Recherche, 95 Rue du Montmurier, 38290 Saint-Quentin Fallavier, France article info abstract

Article history: The evolution of compressive strength belongs to the most fundamental properties of cement paste. Driven by an Received 19 June 2015 increasing demand for clinker substitution, the paper presents a new four-level micromechanical model for the Accepted 8 December 2015 prediction of compressive strength of blended cement pastes. The model assumes that the paste compressive Available online 9 March 2016 strength is governed by apparent tensile strength of the C-S-H globule. The multiscale model takes into account the volume fractions of relevant chemical phases and encompasses a spatial gradient of C-S-H between individual Keywords: grains. The presence of capillary pores, the C-S-H spatial gradient, clinker minerals, SCMs, other hydration prod- Compressive strength (C) Calcium-Silicate-Hydrate (C-S-H) (B) ucts, and air further decrease compressive strength. Calibration on 95 experimental compressive strength values Microstructure (B) shows that the apparent tensile strength of the C-S-H globule yields approx. 320 MPa. Sensitivity analysis reveals Cement paste (D) that the “C-S-H/space” ratio, followed by entrapped or entrained air and the spatial gradient of C-S-H, have the Blended cement (D) largest inﬂuence on compressive strength. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction dealing with blended binders [5]. The main reason lies in altered chem- istry where a chemically heterogeneous gel differs in chemical phases, Compressive strength of concrete and its evolution belong to the e.g. CH may be depleted in pozzolanic reactions or C-A-H phases most important and tested parameters. Due to the multiscale nature may emerge. To overcome these deficiencies, micromechanical models of concrete spanning the range from sub-nanometers to meters and taking into account volume fractions directly have been set up for its composite nature, several factors on different scales play a role in more fundamental understanding. concrete compressive strength [1]. The most relevant initial factors in- Continuum micromechanical models able to reproduce the evolu- clude the binder type, aggregate type, extent of the interfacial transition tion of stiffness and compressive strength have recently been published, zone, and the air content. Time-dependent factors play a role mainly in e.g. for Portland-based materials [6,7] or for cocciopesto mortars [8]. binder’s reaction kinetics which is reflected in the evolution of chemical Pichler et al. [9] used spherical and acicular representation of hydrates phases with a direct impact on stiffness and strength evolution. to capture percolation threshold during hydration and the onset on Up to the time of Hoover Dam construction in the 1930s, a cementi- elasticity and strength. Hydrates consisted of solid C-S-H, small and tious binder was mostly equivalent to Portland cement. Since that time, large gel pores and other hydration products. The strength criterion Portland clinker has been substituted more and more by supplementary was based on deviatoric strength of hydrates which was identified to cementitious materials such as slag, fly ash, limestone or silica fume. The be 69.9 MPa. Once the quadratic stress average in arbitrarily oriented substitution rose from 17% in 1990 to 25% in 2010 among the top world needle-shaped hydrates exceeded this strength, the material failed in cement producers [2]. A further shift is expected, motivated by econom- a brittle manner. Similar modeling approach was also used for C3S, ical, ecological and sustainable benefits [3,4]. C2S, C3S+C3A + gypsum pastes, assuming that only C-S-H with the The question on the origin of concrete strength becomes reinitiated Mohr–Coulomb quasi-brittle failure criterion was responsible for the with the advent of blended binders. The famous Powers' empirical rela- compressive strength of pastes [10]. Computational micromechanical tionship between the gel-space ratio and compressive strength devel- models generally allow taking into account nonlinear elasto-plasto- oped for Portland-based materials needs several adjustments when damage constitutive laws at the expense of computational time. Several 2D and 3D lattice and continuum models were applied for cement paste [11] or concrete [12,13,14] to mention a few. ⁎ Corresponding author. In this paper, a new four-level micromechanical fracture model for E-mail address: [email protected] (M. Hlobil). 1 Gilles Chanvillard (21.1.1963 - 24.10.2015) passed away very unexpectedly during the blended cement pastes is presented, starting from C-S-H globule up to revision process of this paper. cement paste with entrapped/entrained air. The lowest homogenization

level contains C-S-H globules, small and large gel pores [15]. A C-S-H strength, ft. Damage mechanics uses the concept of the equivalent globule is considered to be the only strain-softening component in the strain, εe, as a descriptor of damage evolution. Damage becomes initiated multiscale model, leading essentially to failure at each level. Softening when the equivalent strain, eε, exceeds strain at the onset of cracking, occurs under excessive tension or compression described by an elasto- ε0 =ft/E, where E is the elastic modulus. The Rankine criterion for tensile damage constitutive law introduced in Section 2.1. Higher levels also failure defines εe as use the elasto-damage constitutive law for a phase containing C-S-H σ globules with an updated homogenized stiffness, strength, and estimat- e 1 ε ¼ ; σ 1 N 0 ð1Þ ed fracture energy. Hence, the strength of cement paste origins from the E softening and failure of C-S-H globule. The multiscale model accounts where σ is the maximum positive effective principal stress on for volume fractions of C-S-H, other hydrates, capillary porosity, clinker, 1 undamaged-like material, see Fig. 1 (a). supplementary cementitious materials, entrained/entrapped air and Under uniaxial compressive stress, crack initiation occurs under a considers the spatial gradient of C-S-H among clinker grains. different mechanism. A homogeneous material experiences only one The model contains two independent variables that need to be cali- negative principal stress and the deviatoric stress. Cracking in diagonal, brated from experimental data: the apparent tensile strength of C-S-H shear band zone is often encountered on cementitious specimens, how- globules and the spatial gradient of C-S-H. Numerical results for elastic ever, the physical mechanism is again tensile microcracking in voids modulus and compressive/tensile strength on each scale are further and defects of the underlying microstructure [pp 297, 17]. Such a behav- fitted to microstructure-calibrated analytical expressions, speeding up ior has already been described in the work of Griffith [18], and McClin- the whole validation part. Sensitivity analysis identifies key compo- tock and Walsh [19], and we briefly review this theory and extend it nents for the compressive strength of blended pastes. with an equivalent strain to be used in the framework of damage mechanics. 2. Finite element analysis It is assumed that a material contains randomly oriented 2D elliptical flat voids with various aspect ratios m=b/a,seeFig. 1 (b). Further nota- 2.1. Fracture material model and its implementation tions assume that tensile stress is positive and σ1 ≥σ3.Thevoidshavea negligible area and only represent stress concentrators and internal de- The nonlinear constitutive behavior of quasi-brittle materials, such fects in a material. Under macroscopic biaxial stress, the maximum ten- as cement paste, mortar or concrete, can be generally described by sile stress among all voids, m⋅ση, appears on a critically inclined three common theories: plasticity, fracture mechanics, damage me- elliptical void under a critical angle ψ chanics, or their combinations. Plasticity fails to describe stiffness degradation which is needed for strain softening, strain localization, and the σ 3 σ 1 σ 1 1 size effect, although several extensions were proposed [16]. Linear elas- cos2ψ ¼ ; ≥ ð2Þ 2ðÞσ 3 þ σ 1 σ 3 3 tic fracture mechanics can only deal with brittle materials with a negligible process zone ahead of a crack tip and cannot handle microcrack 2 ðÞσ 1 σ 3 nucleation into a macrocrack. Damage mechanics, particularly the cohe- m σ η ¼ ð3Þ 4ðÞσ þ σ sive crack model, defines the traction–separation law in a plastic zone of 1 3 an opening crack which is related to the fracture energy of a material Crack formation occurs when the tangential tensile stress, m⋅ση,equals and stiffness degradation. This nonlinear fracture mechanics approach to the tensile strength of the matrix. Since ση and the crack geometry, m, offers reasonable stress–strain predictions with minimum parameters, cannot be measured directly, it is reasonable to relate their product to reasonable computation time, and captures the size effect of quasibrittle the uniaxial macroscopic tensile stress, σ , as proposed by Griffith [18] materials [17]. More elaborated plastic-damage models have been suc- 1 cessfully used for the mesoscale analysis of concrete [14]. m σ η σ ¼ ð4Þ The present material model is based on fracture mechanics on all 1 2 considered four scales. A crack starts to grow when cohesive stress is exceeded anywhere in the material. For uniaxial tension of a homoge- The material starts to crack when σ 1 equals to the uniaxial macro- neous material, this cohesive stress corresponds obviously to the tensile scopic tensile strength ft. Note that the tensile strength of the

Tensile stress 1 Compressive stress -| 3|

a b 1 1

m·

Tensile stress 1 Compressive stress -| 3| (a) (b)

Fig. 1. Crack evolution during (a) uniaxial tensile stress and (b) compressive stresses. The material contains randomly oriented elliptical voids with negligible area. 190 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 homogeneous matrix (intrinsic strength) and the crack geometry Rankine model provides in addition the equivalent strain which is remain unknown separately and we can assess only apparent macro- used for damage quantification. scopic tensile strength ft. Plugging Eq. (3) into Eq. (4) and further The Griffith–Rankine model was implemented in OOFEM package into Eq. (1) leads to the definition of the equivalent strain, eε,under [21]. The basic test covered a uniaxially loaded one brick element 2 compression-dominant loading under pure tension or compression for E=20 GPa, Gf =20J/m , ft = 5 MPa and the element size 10×10×10 mm. Fig. 3 shows the response ðÞσ σ 2 εe ¼ 1 1 3 ð Þ in terms of average stress and strain over the element. Note that the lin- 5 fi E 8ðÞσ 1 þ σ 3 ear softening law de ned in the integration point may be distorted on the element scale since damage occurs in more integration points and An interesting feature of the Griffith model is that the ratio of the later becomes localized into one integration point. The material model uniaxial compressive-to-tensile strength equals to 8 provides fracture energy in compression, which is theoretically 64 times larger than that in tension. However, such fracture energy in com- j j¼ ð Þ f c 8f t 6 pression is a fictitious value since other mechanisms for energy dissipation occur such as buckling within the microstructure [22]. fi σ This can be veri ed when plugging 1 =ft into Eq. (1) which leads to To demonstrate further the performance of the Griffith–Rankine εe ¼ = σ σ f t E. Plugging 1 =0, 3 =-8ft into Eq. (5) leads to the same equiv- model, Fig. 4 shows the strain localization in shear bands under the uni- alent strain. axial compression test under plane stress condition. A layer of finite el- Since the equivalent strain may arise from the Rankine of Griffith ements with steel properties is attached to the top and bottom which criterion, it is necessary to compare both Eqs. (1) and (5) and to select restrains the horizontal displacement and induces the horizontal shear the higher equivalent strain. Linear softening is assumed in the simula- stress in the contact. This mimics the situation of the standard uniaxial tions. The softening shape has a little effect on the computed macro- compression test without teflon pads. Material properties remain the scopic strength which is the main objective of this paper and naturally same as in the previous case and the microstructure size 20×20 mm affects the post-peak behavior. The linear cohesive law takes the form is discretized to 20×20 finite elements. Black lines show crack orientation in damaged parts. ω σ ¼ ð Þ f t 1 ω 7 f 2.2. Boundary conditions and intrinsic properties where w is a crack opening and wf is the maximum crack opening at The four-level multiscale model utilizes a 2D or 3D finite element zero stress. According to the formulation of the isotropic damage analysis on each scale, where the Griffith-Rankine constitutive laws model, the uniaxial tensile stress obeys the law govern the crack growth and material softening. The discretization of the microstructure relies on the length scale considered. Each scale con- σ ¼ ðÞ1 ω E εe ð8Þ tains relevant chemical phases in correct volume proportions. The prescribed boundary conditions on a unit cell (UC) reflect uniaxial Let us consider the fracture energy in mode I, G , and the effective f loading on each scale. Periodic boundary conditions would provide the thickness of a crack band h, which corresponds to the finite element most reliable estimation of elastic properties when cement paste is con- size in the direction of the maximum principal strain [17].This sidered as a whole at one homogenization level, without any hierarchical ensures objective results, independent on finite element size [20].No- approach [23]. However, kinematic boundary conditions in the top and ¼ ωεe tice that w h and wf =2Gf / ft. The evolution of isotropic damage bottom UC's surfaces yield a better match for this hierarchical approach is obtained by combining Eqs. (7) and (8) for the elastic modulus. The UC is loaded by eigenstrain increment in

1 the vertical direction which builds up the uniaxial macroscopic compres- ε hEε2 ω ¼ 1 0 1 0 ð9Þ sive stress. εe 2G f Table 1 summarizes intrinsic elastic and apparent fracture properties assigned to relevant phases during simulations. While nanoindentation The performance of the proposed Griffith–Rankine model under the previously quantified elastic properties [24], the tensile strengths of C- plane stress condition is demonstrated in Fig. 2 for the onset of cracking. S-H and C-S-H globules need to be estimated by means of downscaling The Mohr-Coulombp modelffiffiffi gives similar results for the friction angle of from experimental data on cement pastes, see Section 4.6. = ¼ 51.1° and c0 f t 2, which keeps the compressive/tensile strength The fracture energy of individual components with their inter- ratio of 8. The differences in both models are small but the Griffith– faces is estimated, based on available data. For example, the C-S-H– C-S-H debonding energy was computed to lay between 0.38 and

-3 Macroscopic strain • 10 (-) 5 -4 -3 -2 -1 0 1 2 0 -5 -10 -15 -20 -25 -30 -35

-40 Macroscopic stress (MPa)

Fig. 2. Surface at the onset of cracking according to two models under the plane stress Fig. 3. Test of one linear brick element under uniaxial tension/compression with 8 condition. integration points. M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 191

Level 2: Cement paste adds unhydrated clinker and all remaining hydration products (CH, AFm, AFt,...) inside the homogenized C-S-H foam. The characteristic size spans the range of 10 - 100 μm. Level 3: Cement paste with air captures entrained or entrapped air, decreasing the resulting mechanical properties. The characteristic size spans the range of 100-1000 μm. C-S-H foam in Level 1 covers a region between individual clinker grains. The size of the region depends on (i) the mean size of clinker grains, and on (ii) the initial composition of cement paste specified by the water-to-binder mass ratio. In modeling, such microstructure was simplified by assigning a constant UC height of 10 μm. It is worthy to mention that C-S-H is not a true gel but a precipitate, signalized by the presence of capillary porosity, by a more-or-less constant gel porosity during the C-S-H formation, and by the C-S-H Fig. 4. Strain localization in shear bands under the uniaxial compression test under plane stress. 20×20 quadrilateral elements (20×20 mm), deformation exaggerated 40×, appearance mainly around cement clinker grains [31] which can be damage showed for the end of softening. observed in SEM/BSE images. To investigate the spatial distribution of C-S-H between grains, μCT scans of cement pastes from The Visible Cement Dataset created by NIST were used [32]. Two different ce- 2.85 J/m2[25]. However, this energy is too low and the pull-out ment pastes were considered with water/cement ratio equal to 0.3 mechanism of CH crystals increases this energy to a theoretical and 0.45, both produced from a reference cement #133. Cement range of 7.55–283 J/m2, which is more realistic with measured values paste with w/c = 0.3 was investigated at 8 and 162 h after mixing of fracture energy of cement pastes [25]. Molecular dynamics simula- whereas cement paste with w/c = 0.45 was studied at 8 and 124 h. tions performed on atomic scale yield fracture energy of C-S-H as All μCT scans were sliced to create 2D images, thresholded to sepa- 1.72±0.29 J/m2 [26]. The value of fracture energy is not very sensitive rate clinker, hydrates and capillary pores based on known volume parameter for the peak stress of a material, see Fig. 29,butratherinflu- fractions. In this particular case, C-S-H corresponded to hydrates. ences post-peak behaviour which is not the primary focus of this paper. Fig. 6 shows the calculated two-point correlation function between hydrates (C-S-H) and the closest grain surface. Thresholded images of pastes tested 8 hours after casting show that C-S-H precipitate 3. Multiscale model for cement pastes mainly around cement grains for both w/c ratios studied. Older pastes exhibit a denser microstructure but a larger w/c ratio (here The previous success of the gel-space descriptor coined by Powers w/c = 0.45) indicates a higher gradient of C-S-H. indicates that strength emerges mainly on submicrometer resolution A common assumption of hydration models is concentric growth of [7,30]. spherical shells representing hydration products around original grains A four-level hierarchical model for cement paste, see Fig. 5, is con- such as HYMOSTRUC [33,34] or μic [35,36]. In the proposed multiscale structed to account for major and minor effects influencing strength. model we approximate the spatial gradient of the C-S-H distribution The coupling between levels occurs through upscaling the elastic by a spatial density function ρ as modulus, E, and the tensile strength, f . The tensile strength on each t ! fi fi level is identi ed from compressive strength, fc, using the Grif th crite- 1 x=ℓ ρðÞ¼x; β exp UC ð10Þ rion, which yields a constant ratio fc/ft =8,see Section 2.1.Thefracture β β energy, Gf, is not directly upscaled, instead, it is explicitly defined at each level, see further discussion in Section 4.13. Level 0: C-S-H considers C-S-H globules packing to densities be- where x stands for the distance from the clinker grain surface, β is a pa- tween C-S-HLD and C-S-HHD. Together with Jennings [15], the C-S-H rameter of the exponential distribution and ℓUC is the characteristic globule consists of calcium silicate sheets, interlayer space, intraglobule length of UC. Low values of β indicate a high gradient of C-S-H in space, and monolayer water. The globules become mixed with small space. Fig. 7 illustrates the impact of β on the C-S-H distribution at the and large gel pores (1 nm bdb12 nm). The characteristic size spans level of C-S-H foam. the range of 1-100 nm. The term “gel” coined by Powers described a mixture of amorphous Level 1: C-S-H foam contains C-S-H together with capillary porosity. C-S-H and all other crystalline hydration products [30].However,recent

To mimic an uneven distribution of C-S-H among grains in cement detailed modeling of C3SandC2S pastes revealed that not all hydration paste, C-S-H have a spatial gradient. The characteristic size spans the products contribute to the compressive strength equally [10]. In fact, range of 100 nm-10 μm. more precise results for the compressive strength prediction were

Table 1 Intrinsic elastic and apparent fracture properties of relevant chemical phases considered in the multiscale hierarchical simulation.

Elastic modulus Poisson's ratio (−) Tensile (compressive) Fracture energy Ref. (GPa) strength (MPa) (J/m2)

C-S-H globule 57.1 0.27 320 (2560) 2a [10] b a C-S-HLD 21.7 0.24 66 (528) 5 [27] b a C-S-HHD 29.4 0.24 107 (856) 5 [27] Water-ﬁlled porosity 0.001 0.499924 ––[23] CH 38.0 0.305 ––[27] AFt 22.4 0.25 ––[28] AFm 42.3 0.324 ––[28] Clinker minerals 139.9 0.30 ––[7] Mature cement paste –– – 15 [29]

a ) Higher fracture energy leads to the same material strength. b ) Calculated from packing of C-S-H globules. 192 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202

Fig. 5. Four-level hierarchical multiscale model. TEM/SEM images from left to right: Eric Lachowski, S.Y. Hong and F.P. Glasser (http://publish.illinois.edu/concretemicroscopylibrary/ transmission-electron-microscopy-tem/); L. Kopecký (CTU in Prague); Paul Stutzman (NIST, http://publish.illinois.edu/concretemicroscopylibrary/magniﬁcation-progression-series- 1000x/); D. P. Bentz, et al. (Visible Cement Dataset: http://visiblecement.nist.gov) obtained when C-S-H was the only failing component in the cement self-consistent scheme [38] is used to estimate the effective elastic paste. A similar conclusion is supported by Fig. 8 which shows property of this homogenized phase which is based on the volume frac- the contribution of clinker minerals to the overall compressive tions and elastic properties of individual microstructural constituents strength of hydrated pastes [37]. While minerals producing C-S-H according to Table 1. The homogenized phase is then embedded into yield comparable results, C3A generating AFt yields a much lower the C-S-H foam where it acts as solid inclusions. strength. In the presented multiscale model, C-S-H are considered Level 3 describes the effect of entrapped or entrained air inside the as the weakest active binding phase, interconnecting other crystalline cement paste which has a pronounced impact on the mechanical prop- hydration products, unreacted clinker minerals, and SCMs. Instead of erties. The microstructure on this level consists of circular voids which using the “gel/space ratio”,wedeﬁne the term “C‐S‐H/space ratio”, are embedded into a homogeneous cement paste.

γCSH,as

ϕ 4. Results and discussion γ ¼ CSH ð Þ CSH ϕ þ ϕ 11 CSH cappor The four-level model aims to predict the elastic modulus and compressive/tensile strength of cement pastes by hierarchical ho- where ϕCSH and ϕcappor stand for the volume fractions of C-S-H (includ- mogenization. The tensile strength of the C-S-H globule plays the ing gel pores) and capillary porosity, respectively. most important role since it is projected onto higher hierarchical levels. Level 2 of the multiscale model describes cement paste as a two- This strength value will be calibrated from macroscopic data since no phase material. For the reason of simplicity, crystalline hydration prod- direct experimental method is available today, see Section 4.6. The re- ucts, clinker minerals, and SCMs are merged together into one ﬁctitious sults from ﬁnite element simulations are approximated using analytical isotropic elastic phase, see Fig. 5. Due to the crystalline morphology, a functions on all hierarchical levels to speed up the validation.

(a) (b)

Fig. 6. Spatial distribution of C-S-H between grains for a) early-age pastes and b) for older pastes. One pixel corresponds to 0.95 μm. M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 193

Fig. 7. The impact of the parameter β on the C-S-H distribution (Level 1) for C-S-H/space ratio γCSH =0.7, UC size 5×5×10 μm.

4.1. Level 0: C-S-H The apparent tensile strength of the globule governs strength at higher hierarchical levels. Section 4.6 shows that the globule’sapparent The presented level considers time-dependent and spatially uniform tensile strength was identified as packing density of C-S-H globules as an approximation of a colloidal sys- ¼ ð Þ tem of C-S-H gel. The shape of the globule is set as one voxel (5 nm), f t;glob 320 MPa 15 without specifying globule's orientation, see Fig. 10.Theglobuleis treated as an isotropic material. Fig. 10 shows UCs for C-S-HLD and C-S-HHD types used for finite Jennings [39] coined two morphologies of C-S-H (low and high den- element simulations. Fig. 11 displays the evolution of the C-S-H tensile sity) which emerge from different packing densities of C-S-H globules. strength depending on the packing density. The approximation reads Nanoindentation experiments carried out by Constantinides and Ulm [27] identified elastic properties of C-S-H, see Table 1, as well as derived ! 1 η 5:133 η η 0 glob limit packing densities as CSH =0.63 and CSH =0.76 [40]. A packing f ¼ f exp 1:101 ð16Þ LD HD t;CSH t;glob η density of globules is estimated from the CEMHYD3D hydration model glob [41] with the implemented confinement algorithm [23] and it is a function of the initial water-to-binder mass ratio, w/b, and the time- which yields the apparent tensile strength of C-S-HLD and C-S-HHD as

ϕ 0 0 dependent capillary porosity, cappor. Fig. 9 shows the evolution for ft ,CSHLD =66 MPa and ft ,CSHHD =107 MPa, respectively. By extrapolation three w/b ratios with the approximation of volume fraction of C-S-HHD to globule packing density ηglob =1 we identify the apparent tensile within C-S-H as strength of the globule as 320 MPa. The compressive strength of C-S- H can be obtained from the Grifﬁth criterion in Eq. (6), which yields hi 1 ðÞ2:5þ5 w=b the apparent compressive strength fc0 ,CSH =528 MPa, and fc0 ,CSH = ϕ ¼ 1 þ 5000 ϕ ð12Þ LD HD CSHHD cappor 856 MPa.

The conﬁnement algorithm makes no difference between various solid 4.2. Level 1: C-S-H foam phases, hence it captures both OPC and blended cements [23].Depend- ing on the volume fraction of C-S-HHD within C-S-H, we interpolate the C-S-H precipitating in the region between individual clinker grains actual packing density of globules as forms a spatial gradient, see Fig. 7. This level contains only C-S-H which is intermixed with capillary porosity. The computational UC is η ¼ η þ η η ϕ ¼ 0:63 þ 0:13 ϕ ð13Þ composed from a 3D uniform mesh, consisting of 20×20×40 brick ele- glob CSHLD CSHHD CSHLD CSHHD CSHHD ments with linear displacement interpolations and with a resolution of 0.25 μm/voxel, yielding the UC size of 5×5×10 μm. The proposed setup Similarly, Young’s modulus of C-S-H spans the range between assumes that the top and bottom planes of UC touch the surfaces of clin- ECSH =21.7 GPa and ECSH =29.4GPa,see Table 1. LD HD ker grains and C-S-H precipitates in between. C-S-H is generated ran- ÀÁ domly in UC but its spatial distribution follows Eq. (10). Fig. 12 shows E0 ¼ E þ E E ϕ ¼ 21:7 þ 7:7 ϕ ð14Þ CSH CSHLD CSHHD CSHLD CSHHD CSHHD the stress-strain diagram obtained from a FE model of C-S-H foam

Fig. 9. Evolution of the volume fraction of C-S-HHD within C-S-H as a function of capillary

Fig. 8. Compressive strength of pastes with w/c = 0.45 measured on cylinders with a porosity and the initial composition from the CEMHYD3D model. C-S-HLD always appear diameter of 12.7 mm and height of 25.4 mm [37]. at the beginning of hydration. 194 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202

Fig. 12. Stress-strain diagram obtained on C-S-H foam using γCSH =0.7, β=0.6, and

ft0,CSH=66 MPa.

Fig. 10. UCs for C-S-HLD and C-S-HHD, 20×20×40 brick elements (UC size 100×100×200 nm).

consisting solely of C-S-HLD using the C-S-H/space ratio γCSH=0.7,β= 0 β−1 0.6, and f t , CSH =66MPa. C1 ¼ 1:101 exp −0:296 ; β ∈ hi0:4; 1:0 ð21Þ The results from numerical simulations on the level of C-S-H foam β show that the spatial gradient has only a weak impact on elasticity. 1:987 The elastic modulus for percolated microstructures linearly increases D1 ¼ −11:058β þ 16:191β; β ∈ hi0:4; 1:0 ð22Þ with the increasing C‐S‐H/space ratio, see Fig. 13. An analytical approximation to numerical results of the elastic modulus of C-S-H foam yields

4.3. Level 2: cement paste γ B1 − I ¼ 0 CSH 1 ð Þ Efoam ECSH exp A1 γ 17 CSH Level 2 uses a 2D uniform mesh composed of 100×100 quadrilateral elements (UC size of 100×100 μm) with linear displace- ! : ment interpolations. The comparison with 3D UC proved that 2D β0 02−1 A ¼ 0:820 exp −4:949 ; β ∈ hi0:4; 1:0 ð18Þ model under plane stress yields very similar results for elasticity 1 β2:8 and peak stress. The 2D model allows capturing larger and more detailed microstructures. The damage model was assigned to a ho- ! : β0 02−1 mogeneous C-S-H foam. The actual size of inclusions follows a step- B1 ¼ 1:818 exp 4:310 ; β ∈ hi0:4; 1:0 ð19Þ wise particle size distribution with circles of 5, 10, and 15 μmin β2:8 diameter. Introducing a solid inclusion into a homogeneous C-S-H foam results where β ∈ 〈0.4;1.0〉. Higher β values are equivalent to a uniform in an increase of the stiffness of the reinforced foam on the one hand, distribution. and in a decrease of the compressive strength on the other hand. Results Compressive strength determined on identical microstructures ex- from FE simulations show that the overall stiffness of cement paste pro- hibits a high sensitivity on the spatial gradient, see Fig. 14.Thiscanbe portionally increases with the quantity of inclusions as well as their explained by the stress concentration in diluted areas which decrease stiffness, see Fig. 15. the load-bearing capacity. The analytical approximation to numerical An analytical approximation for elasticity prediction yields data results in more complicated equations, capturing the effect of the hi ‐ ‐ γ 0:2 C S H/space ratio CSH and the spatial gradient by means of the param- II ¼ I þ ϕ : þ : I ð Þ ECP Efoam 1 incl 0 0102Eincl 0 278Efoam 23 eter β as

The high stiffness contrast between solid inclusions and the C-S-H foam 1−γ D1 I ¼ 0 − CSH ð Þ inﬂuences stress trajectories to concentrate between inclusions, f c;foam f c;CSH exp C1 γ 20 CSH resulting in a decrease of the nominal strength of cement paste, see

Fig. 13. Elastic modulus evolution for Level 1 (C-S-H foam), assuming solely C-S-HLD with η η Fig. 11. Tensile strength of C-S-H depending on the globule packing density. Circles represent the globule packing density glob = CSHLD =0.63. Points represent results from FE results from FE simulations on 20×20×40 brick elements (100×100×200 nm), β=1.0and simulations on UC 20×20×40 brick elements (UC size 5×5×10 μm) and the lines are 2 Gf=2 J/m . analytical approximations. M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 195

600 5x5x10 μm, β=1.0 500 5x5x10 μm, β=0.6 μ β

(MPa) 5x5x10 m, =0.4 Ι c,foam 400

300

200

100

Compressive strength f 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ φ φ φ CSH/space ratio CSH = CSH / ( CSH+ cappor) (−)

Fig. 14. Compressive strength evolution for Level 1 (C-S-H foam), assuming solely C-S-H LD Fig. 16. Compressive strength evolution for level 2 (cement paste), assuming fcI ,foam = with the globule packing density η =η =0.63 and the corresponding compressive glob CSHLD 50,100,150,200 MPa. Circles represent results from FE simulations on a 2D strength fc0,CSH=528 MPa. Points represent results from FE simulations on UC 20×20×40 microstructure on 100×100 quadrilateral elements (UC size 100×100 μm) for brick elements (UC size 5×5×10 μm) and lines are analytical approximations. fcI , foam =50,100,150,200 GPa with varying stiffness of the inclusion Eincl = 40,50,60,70,105,139.9 GPa.

Voids in a homogeneous matrix cause the concentration of stress, Fig. 16. The impact of inclusion’s stiffness on the resultant decrease of effectively reducing the compressive strength, see Fig. 18. An analytical compressive strength is insigniﬁcant. approximation to numerical simulations reads An analytical approximation for compressive strength prediction 8 > ϕ yields > II − : II air ϕ ≤ : < f c;CP 0 2 f c;CP : for air 0 0025 III 0 0025 ! f ; þ ¼ 1:6452 II I c CP air > II −4:056 ðÞϕ −0:0025 f ; ¼ f ; ½ð0:758 þ ðÞ1−0:758 expðÞ−29:3 ϕ 24Þ > : air ϕ N : c CP c foam incl : 0 8 f c;CP exp ϕ for air 0 0025 air ð26Þ 4.4. Level 3: cement paste with air Fig. 19 shows that introducing only 0.25% of air decreases compressive Level 3 again uses a 2D uniform mesh composed of 50×50 quadri- strength by approximately 20%. This sharp decrease could hardly be lateral elements (UC size 1000×1000 μm). To account for the random measured experimentally since it compares a material without any de- size of entrapped or entrained air bubbles in real cement pastes, the fects such as macro-defect-free cements with a material with only a few size of the bubbles follows a step-wise particle size distribution with voids or cracks. However, going to the region after this initial reduction spheres of 20, 60, and 100 μmindiameter. yields a strength reduction of approx. 5% per 1% of entrained air, which The presence of air in the cement paste signiﬁcantly reduces both is already consistent with the observation on the concrete scale [pp 557, observed mechanical properties. Fig. 17 shows the decrease of the elas- 1]. tic modulus with the increasing volume fraction of the air content ϕair. An analytical approximation for elasticity prediction reads 4.5. Discussion on FE mesh representations

III ¼ II ðÞ− ϕ ðÞ ECPþair ECP 1 3 air 25 To demonstrate representativeness of a mesh, a cement paste from Level 2 was taken with the size 100 × 100 μm. Three sets of microstruc- Every 1% of entrained (or entrapped) air results in a proportional tures were generated using different mesh sizes, yielding 50 × 50, decrease of the elastic modulus by 3%. 100 × 100, and 200 × 200 quadrilateral elements. Five randomly generated microstructures in each set aimed at capturing statistical effect with 20% volume fraction of solid inclusions with stepwise particle size distribution using 5, 10, and 15 μm circular inclusions.

I Fig. 15. Elastic modulus evolution for Level 2 (cement paste) assuming Efoam = II 1,10,20 GPa. Circles represent results from FE simulations on a 2D microstructure on Fig. 17. Elastic modulus evolution for Level 3 (cement paste with air), assuming ECP = 100×100 quadrilateral elements (UC size 100×100 μm) with varying stiffness of the 20 GPa. Circles represent results from FE simulations on a 2D microstructure on 50×50 inclusion Eincl=40,50,60,70,105,139.9 GPa . quadrilateral elements (UC size 1×1 mm). 196 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202

100 Level 2, 100x100 μm

(MPa) 80 c

Compressive strength f 0 50x50 100x100 200x200 Microstructure mesh size (pixels)

Fig. 18. Compressive strength evolution for Level 3 (cement paste with air). Circles Fig. 20. Results from FE simulations on Level 2 (cement paste) containing 20% of solid represent results from FE simulations on a 2D microstructure on 50×50 quadrilateral inclusions for three mesh sizes. elements (UC size 1×1 mm) for fcII,CP=50,100,150,200 MPa. pastes, 5 from slag-blended pastes, 5 from fly ash-blended pastes, 5 from limestone-blended pastes, and 4 from finely ground quartz- Simulations show that the predicted compressive strength of ce- blended pastes. The volume of air is only recorded for 44 microstruc- ment paste is almost insensitive to the mesh size, see Fig. 20.However, tures with an average value of 0.8%. The latter was assigned to the re- increasing mesh size results in a decrease of scatter in predicted com- maining microstructures as a default value. pressive strengths, in accordance with theoretical predictions [42]. During the calibration of our multiscale model, the parameter β was initially fixed to β=1, which signalized a nearly uniform distribution of 4.6. Calibration of ft,glob C-S-H in space as well as the highest possible compressive strength prediction. The apparent tensile strength for the C-S-H globule was identi- The proposed multiscale model contains two independent variables fied using 95 experimental results of compressive strength as which need to be calibrated from experimental data ¼ ð Þ f t;glob 320 MPa 27 • The apparent tensile strength of C-S-H globules, ft,glob. • The spatial gradient of C-S-H described by the parameter β. This strength is the highest value which is only exceeded by 6 out of In this regard, we created the CemBase database version 1.0, con- 95 measured compressive strength values, see Fig. 21 (a). A further taining experimental results from testing cement pastes and mortars. reduction of the model-predicted compressive strength is achieved The data contained in this PostgreSQL database are structured and by decreasing β, i. e. effectively by creating a gradient of C-S-H systematically sorted into individual categories starting from the among cement grains. The value β =0.6 globally fits the strength characterization of raw materials, followed by the mixture composi- databest,however,therangeβ ∈ 〈0.4,1〉 can accommodate almost tion, volume fractions of relevant phases, and measured mechanical all experimental results. Pastes from blended cements do not exhibit properties. Individual categories are cross-linked to remove any any substantial variations from the trend described by the C-S-H/ duplicities. space ratio, we, therefore, expect that the C-S-H formed from SCMs ex- Up to date, CemBase contains mechanical tests of 399 microstruc- hibits similar mechanical properties when compared to the C-S-H from tures: 297 pastes and 102 mortars for water-to-binder ratios between pure OPC. 0.157 and 0.68. Complete volume fractions are not available for The theoretical tensile strength for cohesive brittle material is every microstructure due to missing experimental data. Whenever related to the E modulus through surface energy. Typical estimates possible, the quantities of phases were calculated using a CEMHYD3D are ft ∈ (0.01− 0.10)E [43,44]. This gives a theoretical tensile hydration model [41] to complement missing microstructural strength of the C-S-H globule in the range ft ∈ (571− 5710)MPa information. which already shows high values on the small scale. Our apparent Compressive strength is recorded for 353 microstructures between 1 tensile strength for the C-S-H globule lies at the bottom range and 750 days, however, our multiscale model was only validated against which already signalizes defects on this small scale. 95 paste compressive strength values as sufficient microstructural com- Recently, tensile strength was measured indirectly on cantilever positions are available here. Out of 95 strength values, 76 are from OPC beams 20 μm long by means of bending [45]. The beams were prepared from 7 year old cement paste via FIB technique. Tensile strength of outer

product (mainly C‐S‐HLD) corresponded to 264.1±73.4 MPa, inner product (mainly C‐S‐HHD) 700.2±198.5 MPa, while our model predicts values of 66 and 107 MPa, respectively. Such strength underestimation probably stems from oversimpliﬁed transition between levels 1 and 2. The homogenized C-S-H foam matrix on level 2 takes over uniform properties from level 1 while in reality C-S-H agglomerate dominantly around clinker grains. Such uneven distribution creates weaker links in the microstructure reducing its strength. Introducing random ﬁeld for C-S-H foam's tensile strength offers a solution how to calibrate multiscale model for matching C-S-H experimental tensile strength. This clearly demonstrates increase of strength with size decrease and supports the high tensile strength of C-S-H globule. The formation of the C-S-H gradient inevitably leads to a decrease Fig. 19. Reduction of cement paste compressive strength with respect to entrapped or of compressive strength. A possible way of interpreting the C-S-H entrained air. gradient is that as C-S-H precipitates on clinker grains it starts to M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 197

(a) (b) (c)

Fig. 21. Validation of model-predicted compressive strength with experiments for β=1.0(a), β=0.6 (b), and β=0.4(c). grow onwards, filling capillary porosity. For cement pastes with a combination of XRD/Rietveld together with thermal gravimetric analy- relatively high w/b ratio clinker grains are widely spaced apart sis and volume balance equations and provides the necessary volume frac- from each other, C-S-H fail to interconnect neighbouring grains to tions for the model. A high water-to-binder ratio creates a sparsely filled create a solid structural network. Densifying the microstructure, ei- microstructure which, at the same time, results in a high gradient of C-S-H. ther by lowering w/b ratio, by providing additional mineral surfaces Fig. 24 (b) shows a validation where a low parameter β=0.6fits the for C-S-H to precipitate onto or by filling capillary porosity by a fine best experimental results. The last experimental strength value mea- material (such as silica fume), inevitably leads to a more dispersed sured at 91 days shows no increase when compared to 28 days. In this C-S-H precipitation. This results in a low gradient of C-S-H (de- particular case the specimen had dried out, which is further supported scribed by a high parameter β) and corresponds to high strength pre- by mass measurements taken directly after demoulding of the specimen dictions, see Fig. 22 (a) for pastes with w/b≤0.4. Pastes with a higher and before testing. The specimen was kept for 91 days in a cooling water-to-binder ratio (w/bN0.4) tend to exhibit a larger gradient of chamber with 95% RH, 20 ∘ C and was wrapped in a food-preservation C-S-H (described by a low parameter β) due to a less dense packing plastic foil to further reduce water evaporation. However, such precau- of clinker grains, see Fig. 22 (b). tions were insufficient and surface microcracks were very likely formed The scaling parameter β takes into account several factors, no- which resulted in a lower-than-expected compressive strength. Table 2 tably heterogeneous nucleation and precipitation of C-S-H and shows a decrease of model-predicted tensile and compressive strength cement fineness. These factors are normally overlooked since with upscaling. calibration data usually come from similar cements without seeding agents or with various blend levels. Recognizing this ef- 4.9. Example #2—Portland cement paste w/c = 0.247 fect opens a way to optimize distribution of C-S-H and to improve a binder. To study the impact of the water-to-binder ratio on observed mechanical properties, we extract a substoichiometric cement paste with 4.7. Validation of elastic modulus w/b=0.247 from CemBase. Volume fractions were calculated by CEMHYD3D hydration model [41] using raw material characterization The elastic modulus of cement paste was validated against 27 found in [46]. The necessary input for the multiscale model comes experimental results from CemBase. The validation assumed β=1.0 from the volume fraction in Fig. 25 (a). A relatively low w/b=0.247in- and β=0.6 for all experimental results, see Fig. 23. Lower values of β dicates a dense microstructure, forcing C-S-H to evenly precipitate with- lead to slightly lower model predictions. in capillary pores. This is best described by β=0.8,seeFig. 25 (b). Table 3 shows a decrease of model-predicted tensile and compressive 4.8. Example #1 - Portland cement paste w/c = 0.51 strength with upscaling.

This example illustrates the performance of the multiscale model on 4.10. Example #3—slag-blended cement paste w/b = 0.53 OPC with w/b=0.51. The microstructural characterization in Fig. 24 (a) was performed at 3, 7, 28, and 91 days at Lafarge Centre de The results from a microstructural characterization carried out on a Recherche according to the test protocol described in [10] using a slag-blended paste with w/b=0.53 by mass (w/b=1.6 by volume),

(a) (b)

Fig. 22. Impact of the water-to-binder ratio on the C-S-H gradient, represented by the parameter β: (a) dense microstructures with w/b≤0.4 exhibit a more uniform C-S-H formation signalized by β=1, whereas (b) sparse microstructures with w/bN0.4 indicate a higher C-S-H gradient. 198 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202

Table 2 Decrease of strength with upscaling for OPC paste, w/b=0.51 at 28 days (degree of OPC

hydration ξ=0.836) using microstructural characterisation as follows: γCSH=0.60742,

fcappor=0.290, fincl=0.2381,Eincl=48.032 GPa,fair=0.008, β=0.6,andft,glob=320 MPa.

Level 0 Level 1 Level 2 Level 3 C–S–H C–S–H Cement Cement paste foam paste with air

i ft 68.754 8.592 6.515 4.730 MPa i fc 550.029 68.738 52.119 37.835 MPa

The presented multiscale model uses four hierarchical levels to Fig. 23. Model predicted elastic modulus for β=1.0 and β=0.6. describe the microstructure. To simplify the strength prediction into a single equation, we make the following assumptions: (i) at Level 0 we

assume the ratio of C-S-H LD:C-S-H HD equal to 4:1; (ii) at Level 1 the β the 45% cement replacement level are shown in Fig. 26 (a). The charac- spatial gradient of C-S-H by =0.6; (iii) at Level 2 the volume fraction terization of individual chemical phases was performed at 1, 3, 7, 28, of inclusions equals to 0.15; and (iv) at Level 3 the volume of entrained and 91 days in the same way as in Section 4.8. The model-predicted air equals to 0.008. The expression of compressive strength of then sim- fi compressive strength is shown in Fig. 26 (b). The most appropriate pli es into a single exponential equation value is β=0.5. 1−γ 5:7072 4.11. Example #4—fly ash-blended cement paste w/b = 0.60 ¼ − : CSH ð Þ f c;paste 320 exp 1 3412 γ 29 CSH The results from a microstructural characterization carried out on a fly ash-blended paste with w/b=0.60bymass(w/b=1.6byvolume), the 45% cement replacement level are shown in Fig. 27 (a). The charac- Fig. 28 shows the comparison of both approaches with very similar re- β terization of individual chemical phases was performed at 1, 3, 7, 28, sults. However, varying the scaling parameter results in upper and and 91 days in the same way as in Section 4.8 . The model-predicted lower bounds that can cover almost all experimental data points. compressive strength is shown in Fig. 27 (b). The most appropriate value is β=0.5. 4.13. Impact of C-S-H fracture energy 4.12. Comparison with Powers' approach It is an interesting feature of the four-level model that compressive or tensile strength always decreases on upscaling. Fracture energy Direct comparison of Powers' gel/space model [47] and the present- may increase on upscaling if other mechanisms occur during the crack ed model is not straightforward because both are based on different mi- propagation, e.g. crack arresting, shielding, or pulling out an inclusion. crostructural descriptors. However, we will show that under some It is also known from concrete experiments that fracture energy in- reasonable assumptions both approaches yield comparable results. creases with the size of aggregates [48]. Powers coined an empirical relation between the microstructure of A direct experimental characterization of fracture energy for C-S-H is pure cement pastes and their compressive strength using “gel/space” currently, up to our best knowledge, beyond experimental evidence. C- ratio (GSR) as a microstructural descriptor. This relation can be expressed S-H as an amorphous precipitate contains micro-crystalline seeds of CH by a power law function as and other crystalline hydration products (AFm, AFt) which increase the observed fracture energy. Modeling of frictional pull-out tests [25] of CH ¼ b ð Þ f c;paste f 0 GSR 28 from the C-S-H matrix tend to increase the fracture energy due to inter- molecular interactions between C-S-H globules. where f0 stands for the intrinsic compressive strength of the material Numerical modeling on Level 1, see Fig. 29, shows that the fracture and b for the best fitcoefficient. Similar power law function can also energy of C-S-H only has an insignificant impact on the predicted com- 0 2 be identified when the microstructure is described by “C-S-H/space” pressive strength. Decreasing Gf , CSH below 2 J/m would increase the ratio, γCSH,seeFig. 28. apparent tensile strength of a globule, ft,glob, [25].

(a) (b)

Fig. 24. Volume fractions for OPC paste with w/b=0.51 (a). Experimentally measured compressive strength (b) compared with multiscale model predictions. M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 199

(a) (b)

Fig. 25. Volume fractions for OPC paste with w/b=0.247 (a). Experimentally measured compressive strength (b) compared to model predictions.

(a) (b)

Fig. 26. Volume fractions for slag-blended paste with w/b=0.53 and the replacement level of 45% by volume (a). Experimentally measured compressive strength (b) compared to model predictions.

(a) (b)

Fig. 27. Volume fractions for ﬂy ash-blended paste with w/b=0.60 and the replacement level of 45% by volume (a). Experimentally measured compressive strength (b) compared to model predictions

Fig. 29. Impact of fracture energy of C-S-H on compressive strength of C-S-H foam. FE Fig. 28. Comparison of modeling approaches: Powers' power law ﬁt and proposed model. model from 10×10×20 brick elements (UC size 5×5×10 μm) using β=1.0. 200 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202

Table 3 Decrease of strength with upscaling for OPC paste, w/b=0.247 at 28 days (degree of OPC

hydration ξ=0.515) using microstructural characterisation as follows: γCSH=0.77564,

fcappor =0.1146, fincl =0.4892, Eincl =79.033 GPa, fair =0.041, β=0.8, and ft,glob =320 MPa.

Level 0 Level 1 Level 2 Level 3 C–S–H C–S–H Cement Cement paste with foam paste air

i ft 79.673 24.404 18.498 9.289 MPa i fc 637.381 195.231 147.985 74.311 MPa

to any homogenization level in particular, see Fig. 5, but covers the Fig. 30. Void size-dependent compressive strength of cement paste obeying the size-effect law. whole range of sizes. The model consists of a homogeneous matrix 2 (fc =100 MPa and Gf =15 J/m ) with voids. The actual void diameter is kept constant at 10 elements whereas the volume fraction changes 4.14. Impact of void size on strength from 0 up to 10%. Changing the element resolution creates different sizes of the identical microstructure, starting from 1×1 μm with void Voids in cement paste occur on different length scales either as cap- size 0.1 μm up to 2×2 cm with void size 2 mm. It is clear that void di- illary pores (size 10 nm−10 μm) or as entrained air (10 μm−500 μm) ameters dN50 μm reduce the strength more than only by their volume or entrapped air (N1 mm). In quasi-brittle materials the size of voids amount. This is due to a more brittle behavior of the cement paste than matters as the scaling of compressive (tensile) strength obeys the the foam. The presence of entrapped air with dN1mmisthemost size-effect law [17] detrimental.

−1 4.15. Global sensitivity analysis by scatter plots 0 d 2 σ ; ¼ Bf 1 þ ð30Þ N c c d 0 Sensitivity analysis by a sampling-based method identifies the key global factors influencing compressive/tensile strength. Compressive where d is the void diameter, d0 ,B,f'c are constants for identification. In strength is calculated using a full factorial design from a combination a perfectly ductile material, the diameter of a void does not matter. The of all input variables on each scale. Model inputs for each level were level of C-S-H foam resembles this situation with the size up to chosen by uniform sampling over all admissible model input as follows: 10 μm, see Fig. 12, which renders the foam less brittle with larger • C-S-H/space ratio γ ∈〈0.3,1〉 for Level 1. softening part. Capillary pores residing in the foam influence strength CSH • Volume fraction of C-S-H HDwithin C-S-H:ϕCSH ∈〈0,1〉 for Level 1. by their volume only, while the shape and their distribution have a little HD • Parameter for C-S-H spatial gradient: β∈〈0.4,1〉 for Level 1. influence. This is the reason why the total amount of capillary porosity • Volume fraction of inclusion: ϕ ∈〈0,1〉 for Level 2. controls compressive/tensile strength, but neither their pore distribution incl • Volume fraction of entrapped/entrained air:ϕ ∈〈0,0.1〉 for Level 3. nor their shape. air The situation changes for entrapped/entrained air voids which have Compressive strength is plotted against a combination of individual a much larger size than capillary voids. In a perfectly brittle material, input variables in a scatter plot which provides a visual indication of the increasing a void size four-times would decrease the strength twice sensitivity of a given input value. Spearman's rank correlation coeffi- according to linear elastic fracture mechanics. Also, the shape of void cient ρ is used to quantify sensitivity; the closer the |ρ|isto1,the influences the strength of material. Cement paste is a quasibrittle more monotonically related the observed variables are. material hence the scaling of strength occurs between ductile and Input variables on all levels cover a wide range of possible combina- brittle asymptotic limits. tions of admissible data for the multiscale model. Fig. 31 shows the most Fig. 30 shows results from a size effect study on a 2D microstructure correlated input variables globally; the C-S-H/space ratio and the vol- (100×100 quadrilateral elements). The model is not directly associated ume fraction of entrapped/entrained air.

(a) (b)

Fig. 31. Important results from global sensitivity analysis; (a) correlation between the compressive strength of C-S-H foam and the C‐S‐H/space ratio, and (b) compressive strength of cement paste and volume fraction of air. M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 201

Table 4 strength against 95 experimental values. Although there is no direct Local sensitivity analysis on OPC paste, w/c=0.51at28days. experimental evidence of such high tensile strength, the theoretical

Model input Xi Default value X Impact on compressive limit value can be as high as 5700 MPa [43,44]. More experimental ∂ f III c;CPþair j and bottom-up evidence is needed for calibrating this apparent ten- strength ∂ Xi X sile strength. ϕ − Air volume fraction air 0.008 491.67 4. The C-S-H/space ratio was found to be the most fundamental param- C-S-H/space ratio γ 0.608 181.85 CSH eter for compressive/tensile strength, in accordance with Powers Spatial gradient of C-S-H β 0.600 58.37 ϕ [30]. The volume of entrained/entrapped air and the spatial gradient C-S-H HD volume fraction CSHHD 0.094 19.16 Inclusion volume fraction ϕincl 0.267 −0.15 of C-S-H are further factors governing the compressive strength of Elastic modulus of inclusion Eincl (GPa) 48.032 0.00 blended pastes. 5. Precipitating C-S-H around grains exhibit a spatial gradient which has a strong impact on the compressive strength of cement paste. 4.16. Local sensitivity analysis The densification of the microstructure due to a decrease of the initial water-to-binder ratio or by progressing hydration leads to a more To further highlight the impact of individual input parameters on uniform distribution of C-S-H. This assumption was proved by true model predictions, local sensitivity analysis was carried out for selected μ CT using two-point correlation functions. cement pastes. Local sensitivity analysis considers the effect of small 6. The multiscale model needs further elaboration for systems beyond perturbations of one model input variable (Xi) to the compressive the OPC basis producing C-S-H. Particularly alkali-activated systems strength while keeping the other inputs constant (X) (N-A-S-H gel), calcium aluminate cements (C-A-H gel), oxychloride, phosphatic cements, etc. ∂Y 7. The multiscale model captures the strain-softening behavior on each ð31Þ ∂ Xi X scale and handles energy dissipation by upscaling E, ft and assigning Gf. This ensures a correct localization limiter on each scale and the transfer of microstructurally-based fracture properties between The partial derivative in Eq. (31) is solved using a forward differ- scales [49]. ence method with the step length equal to 1 % of the input value. The higher the absolute value of the partial derivation, the more sensitive the input parameter. However, the step length is kept equal for as- Acknowledgment sessment of all input variables, disregarding the actual range for the input parameter. This results in unequal model predictions be- The authors gratefully acknowledge “NANOCEM” funding within cause changing the volume fraction of air by 1 % (with a range from Core Project 10 “Micromechanical Analysis of Blended Cement-Based 0 to 0.1) reduces predicted compressive strength significantly more Composites” and support from CTU grant SGS15/030/OHK1/1T/11 and than changing, e.g. β by 1 % (with a range from 0.4 to 1). Therefore from Czech Science Foundation 16-20008S. We appreciate stimulating the presented local sensitivity analysis only shows trends and high- discussions on NANOCEM events, particularly contributions from lights key microstructural factors rather than comparing absolute Bernhard Pichler (TU Wien) and Klaus-Alexander Rieder (WR Grace). values of the partial derivation. NANOCEM’s funding further allowed Michal Hlobil’sinternshipin Local sensitivity analysis was performed on all mixes in four exam- Lafarge Centre de Recherche during 2012–2013. ples (OPC with w/b=0.51, OPC with w/b=0.247, slag-blended paste The authors would like to pay tribute to Gilles Chanvillard, who with 45% clinker replacement, and fly ash-blended paste with 45% clin- passed away on Oct 24, 2015. Here, Gilles contributed significantly to ker replacement). The results were very similar and only the OPC paste microstructural and mechanical understanding of cementitious com- w/b=0.51isshownin Table 4. Results highlighted that apart from the posites, built several bridges between industrial and academic environ- C-S-H/space ratio and entrapped air, the spatial gradient of C-S-H ment and his cheerful spirit helped to enjoy research and science. plays a key role for the compressive strength and elastic modulus of cement paste. References

[1] A.M. Neville, Properties of Concrete, John Wiley & Sons, Inc., 1997 5. Conclusions [2] The Cement Sustainability Initiative, 10 Years of Progress and Moving on into the Next Decade, 2012. The paper describes a four-level micromechanical model for [3] E. Gartner, Industrially interesting approaches to “low-CO2” cements, Cem. Concr. Res. 34 (9) (2004) 1489–1498, http://dx.doi.org/10.1016/j.cemconres.2004.01.021. blended cement pastes aimed at predicting compressive strength. [4] E.M. Gartner, D.E. Macphee, A physico-chemical basis for novel cementitious binders, Cem. The multiscale model uses a newly-proposed “C-S-H/space” ratio as Concr. Res. 41 (7) (2011) 736–749. [5] L. Lam, Y. Wong, C. Poon, Degree of hydration and gel/space ratio of high-volume fly ash/ the primary microstructural descriptor which is applicable for pure cement systems, Cem. Concr. Res. 30 (5) (2000) 747–756, http://dx.doi.org/10.1016/ as well as blended cements. The model introduces a spatial C-S-H S0008-8846(00)00213-1. gradient, mimicking nonuniform distribution between individual [6] B. Pichler, C. Hellmich, Upscaling quasi-brittle strength of cement paste and mortar: a – fi multi-scale engineering mechanics model, Cem. Concr. Res. 41 (5) (2011) 467 476, grains. The most relevant ndings and conclusions can be summa- http://dx.doi.org/10.1016/j.cemconres.2011.01.010. rized as follows: [7] B. Pichler, C. Hellmich, J. Eberhardsteiner, J. Wasserbauer, P. Termkhajornkit, R. Barbarulo, G. Chanvillard, Effect of gel–space ratio and microstructure on strength of hydrating ce- 1. The C-S-H globule was taken as the only failing phase which obeys an mentitious materials: an engineering micromechanics approach, Cem. Concr. Res. 45 (0) (2013) 55–68. elasto-damage law with linear softening. Other chemical phases [8] V. Nežerka, J. Zeman, A micromechanics-based model for stiffness and strength estimation were treated elastically. Validation proved that this assumption can of cocciopesto mortars, Acta Polytech. 52 (6) (2012) 29–37. cover all experimental data on OPC and blended pastes with the ad- [9] B. Pichler, S. Scheiner, C. Hellmich, From micron-sized needle-shaped hydrates to meter- fl sized shotcrete tunnel shells: micromechanical upscaling of stiffness and strength of hy- dition of slag, y ash, limestone and quartz and w/b in the range of drating shotcrete, Acta Geotech. 3 (4) (2008) 273–294. 0.157 up to 0.68 from approximately 1 day of hydration under [10] P. Termkhajornkit, Q.H. Vu, R. Barbarulo, S. Daronnat, G. Chanvillard, Dependence of com- – — room temperature. pressive strength on phase assemblage in cement pastes: Beyond gel space ratio Exper- imental evidence and micromechanical modeling, Cem. Concr. Res. 56 (0) (2014) 1–11, 2. The CemBase database version 1.0 was assembled for validating the http://dx.doi.org/10.1016/j.cemconres.2013.10.007. multiscale model. The database now contains 399 experimental data. [11] F. Ganneau, F.-J. Ulm, J. Gondzio, E. Garboczi, An algorithm for computing the compressive fi strength of heterogeneous cohesive-frictional materials—application to cement paste, 3. The apparent tensile strength of the C-S-H globule was identi ed as computers and geotechnics, Chemo Mech. Interact. Geomater. 34 (4) (2007) 254–266, ft,glob =320 MPa by evaluating the model predicted compressive http://dx.doi.org/10.1016/j.compgeo.2007.02.005. 202 M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202

[12] G. Cusatis, Z.P. Bažant, L. Cedolin, Confinement-shear lattice model for concrete damage in [31] G.W. Scherer, Structure and properties of gels, Cem. Concr. Res. 29 (8) (1999) 1149–1157, tension and compression: II. Computation and validation, J. Eng. Mech. 129 (12) (2003) http://dx.doi.org/10.1016/S0008-8846(99)00003-4. 1449–1458. [32] D.P. Bentz, S. Mizell, S. Satterfield, J. Devaney, W. George, P. Ketcham, J. Graham, J. [13] P. Grassl, M. Jirásek, Meso-scale approach to modelling the fracture process zone of con- Porterfield, The visible cement data set, J. Res. Natl. Inst. Stand. Technol. 107 (2) (2002) crete subjected to uniaxial tension, Int. J. Solids Struct. 47 (7–8) (2010) 957–968, http:// 137–148http://visiblecement.nist.gov. dx.doi.org/10.1016/j.ijsolstr.2009.12.010. [33] K. van Breugel, Numerical simulation of hydration and microstructural development in [14] S.-M. Kim, R.K.A. Al-Rub, Meso-scale computational modeling of the plastic-damage re- hardening cement-based materials, Cem. Concr. Res. 25 (2) (1995) 319–331. sponse of cementitious composites, Cem. Concr. Res. 41 (3) (2011) 339–358. [34] A. Princigallo, P. Lura, K. van Breugel, G. Levita, Early development of properties in a cement [15] H.M. Jennings, Refinements to colloid model of C-S-H in cement: CM-II, Cem. Concr. Res. paste: a numerical and experimental study, Cem. Concr. Res. 33 (7) (2003) 1013–1020. 38 (3) (2008) 275–289. [35] S. Bishnoi, K.L. Scrivener, uic: A new platform for modelling the hydration of cements, [16] M. Jirásek, S. Rolshoven, Localization properties of strain-softening gradient plasticity Cem. Concr. Res. 39 (4) (2009) 266–274, http://dx.doi.org/10.1016/j.cemconres.2008.12. models. Part I: Strain-gradient theories, Int. J. Solids Struct.0 46 (11–12) (2009) 002 URL http://www.sciencedirect.com/science/article/pii/S0008884608002214. 2225–2238, http://dx.doi.org/10.1016/j.ijsolstr.2008.12.016. [36] S. Bishnoi, K.L. Scrivener, Studying nucleation and growth kinetics of alite hydration using [17] Z.P. Bažant, J. Planas, Fracture and Size Effect in Concrete and Other Quasibrittle Materials, uic, Cem. Concr. Res. 39 (10) (2009) 849–860, http://dx.doi.org/10.1016/j.cemconres. CRC Press, 1998. 2009.07.004 URL http://www.sciencedirect.com/science/article/pii/S0008884609001641. [18] A. Griffith, Theory of rupture, in: C. Biezeno, J. Burgers (Eds.),First International Congress [37] J. Beaudoin, V. Ramachandran, A new perspective on the hydration characteristics of ce- for Applied Mechanics, Delft 1924, pp. 55–63. ment phases, Cem. Concr. Res. 22 (4) (1992) 689–694, http://dx.doi.org/10.1016/0008- [19] F. McClintock, J. Walsh, Friction of Griffith cracks in rock under pressure, Proc. Fourth U.S. 8846(92)90021-M. National Congress of Applied Mechanics, New York City 1962, pp. 1015–1021. [38] A.V. Hershey, The elasticity of an isotropic aggregate of anisotropic cubic crystals, J. Appl. [20] M. Jirásek, Z.P. Bažant, Inelastic Analysis of Structures, John Wiley & Sons, Ltd., 2002 Mech. 21 (3) (1954) 236–240. [21] B. Patzák, et al., Object Oriented Finite Element Method—OOFEM, http://www.oofem.org [39] H.M. Jennings, A model for the microstructure of calcium silicate hydrate in cement paste, (since 1993. Cem. Concr. Res. 30 (6) (2000) 101–116. [22] Z.P. Bažant, Y. Xiang, Size effect in compression fracture: splitting crack band propagation, [40] G. Constantinides, F.-J. Ulm, The nanogranular nature of C-S-H, J. Mech. Phys. Solids 55 J. Eng. Mech. 123 (2) (1997) 162–172. (2007) 64–90. [23] V. Šmilauer, Z. Bittnar, Microstructure-based micromechanical prediction of elastic proper- [41] D.P. Bentz, CEMHYD3D: A Three-Dimensional Cement Hydration and Microstructure De- ties in hydrating cement paste, Cem. Concr. Res. 36 (9) (2006) 1708–1718. velopment Modeling Package. Version 3.0. Tech. rep., NIST Building and Fire Research Lab- [24] O. Bernard, F.-J. Ulm, E. Lemarchand, A multiscale micromechanics-hydration model for oratory, Gaithersburg, Maryland, 2005. the early-age elastic properties of cement-based materials, Cem. Concr. Res. 33 (9) [42] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the repre- (2003) 1293–1309. sentative volume element for random composites: statistical and numerical approach, Int. [25] T.T. Ghebrab, P. Soroushian, Mechanical properties of hydrated cement paste: develop- J. Solids Struct. 40 (2003) 3647–3679. ment of structure–property relationships, Int. J. Concr. Struct. Mater. 41 (1) (2010) 37–43. [43] D.K. Felbeck, A.G. Atkins, Strength and Fracture of Engineering Solids, Prentice-Hall, New [26] M. Bauchy, H. Laubie, M.A. Qomi, C. Hoover, F.-J. Ulm, R.-M. Pellenq, Fracture toughness of Jersey, 1984. calcium-silicate-hydrate from molecular dynamics simulations, J. Non-Cryst. Solids 419 [44] H. Gao, Application of fracture mechanics concepts to hierarchical biomechanics of bone (0) (2015) 58–64, http://dx.doi.org/10.1016/j.jnoncrysol.2015.03.031 URL http://www. and bone-like materials, Int. J. Fract. 138 (1-4) (2006) 101–137. sciencedirect.com/science/article/pii/S0022309315001362. [45] J. Němeček, V. Králík, V. Šmilauer, L. Polívka, A. Jäger, Tensile strength of hydrated cement [27] G. Constantinides, F.-J. Ulm, The effect of two types of C-S-H on the elasticity of cement- paste phases assessed by micro-bending tests and nanoindentation, Cem. Concr. Compos. based materials: results from nanoindentation and micromechanical modeling, Cem. (2015) submitted. Concr. Res. 34 (1) (2004) 67–80. [46] J.-M. Mechling, A. Lecomte, C. Diliberto, Relation between cement composition and com- [28] C.-J. Haecker, E.J. Garboczi, J.W. Bullard, R.B. Bohn, Z. Sun, S.P. Shah, T. Voigt, Modeling the pressive strength of pure pastes, Cem. Concr. Compos. 31 (4) (2009) 255–262, http://dx. linear elastic properties of Portland cement paste, Cem. Concr. Research 35 (10) (2005) doi.org/10.1016/j.cemconcomp.2009.02.009. 1948–1960. [47] T.C. Powers, Structure and physical properties of hardened Portland cement paste, J. Am. [29] V. Šmilauer, P. Hlaváček, P. Padevět, Micromechanical analysis of cement paste with car- Ceram. Soc. 41 (1) (1958) 1–6. bon nanotubes, Acta Polytech. 52 (1998) 35–41. [48] F.H. Wittmann, Crack formation and fracture energy of normal and high strength concrete, [30] T.C. Powers, T.L. Brownyards, Studies of physical properties of hardened portland cement Sadhana 27 (4) (2002) 413–423. paste, Bulletin 22, Research Laboratories of the Portland Cement Association, Chicago, [49] Z.P. Bažant, Can multiscale-multiphysics methods predict softening damage and structural March 1948. failure? Int. J. Multiscale Comput. Eng. 8 (1) (2010) 61–67.